Abstract
It is shown that for everyk and everyp≥q≥d+1 there is ac=c(k,p,q,d)<∞ such that the following holds. For every familyℋ whose members are unions of at mostk compact convex sets inR d in which any set ofp members of the family contains a subset of cardinalityq with a nonempty intersection there is a set of at mostc points inR d that intersects each member ofℋ. It is also shown that for everyp≥q≥d+1 there is aC=C(p,q,d)<∞ such that, for every family
of compact, convex sets inR d so that among andp of them someq have a common hyperplane transversal, there is a set of at mostC hyperplanes that together meet all the members of
.
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N. Alon, I. Bárány, Z. Füredi, and D. J. Kleitman, Point selections and weak ε-nets for convex hulls,Combin. Probab. Comput. 1 (1992), 189–200.
N. Alon and G. Kalai, A simple proof of the upper bound theorem,European J. Combin. 6 (1985), 211–214.
N. Alon and D. J. Kleitman, Piercing convex sets and the Hadwiger Debrunner (p, q)-problemAdv. in Math. 96 (1992), 103–112. See also: Piercing convex sets (research announcement),Bull. Adv. in Math. Soc. 27 (1992), 252–256.
I. Bárány, A generalization of Caratheodory's theorem,Discrete Math.,40 (1982), 141–152.
S. Cappell, J. Goodman, J. Pach, R. Pollack, M. Sharir, and Rephael Wenger, Common tangents and common transversals,Adv. in Math.,106 (1994), 198–215.
B. Chazelle and J. Friedman, A deterministic view of random sampling and its use in geometry,Combinatorica 10 (1990), 229–249.
L. Danzer, B. Grünbaum, and V. Klee,Helly's Theorem and Its Relatives, Proceedings of Symposium in Pure Mathematics, Vol. 7 (Convexity), American Mathematical Society, Providence, RI, 1963, pp. 101–180.
V. L. Dol'nikov, A coloring problem,Siberian Math. J 13 (1982), 886–894; translated fromSibirsk. Mat. Zh. 13 (1972), 1272–1283.
J. Eckhoff, An upper bound theorem for families of convex sets,Geom. Dedicata 19 (1985), 217–227.
J. Eckhoff, Helly, Radon, and Carathèodory type theorems, inHandbook of Convex Geometry (P. Grüber and J. Wills, eds.), North-Holland, Amsterdam, 1993, pp. 389–448.
J. Eckhoff, A Gallai type transversal problem in the plane,Discrete Comput. Geom. 9 (1993), 203–214.
B. E. Fullbright, Intersectional properties of certain families of compact convex sets,Pacific J. Math. 50 (1974), 57–62.
B. Grünbaum, On intersections of similar sets,Portugal Math.,18 (1959), 155–164.
B. Grünbaum, Lectures on Combinatorial Geometry, Mimeographed Notes, University of Washington, Seattle, 1974.
B. Grünbaum and T. Motzkin, On components in some families of sets,Proc. Amer. Math. Soc. 12 (1961), 607–613.
H. Hadwiger and H. Debrunner, Über eine Variante zum Helly'schen Satz,Arch. Math. 8 (1957), 309–313.
H. Hadwiger, H. Debrunner, and V. Klee,Combinatorial Geometry in the Plane, Holt, Rinehart, and Winston, New York, 1964.
E. Helly, Über Mengen konvexer Körper mit gemeinschaftlichen Punkten,Jahresber. Deusch. Math.-Verein. 32 (1923), 175–176.
G. Kalai, Intersection patterns of convex sets,Israel J. Math. 48 (1984), 161–174.
M. Katchalski and A. Liu, A problem of geometry inR d,Proc. Amer. Math. Soc. 75 (1979), 284–288.
H. Morris, Two pigeonhole principles and unions of convexly disjoint sets, Ph.D. thesis, California Institute of Technology, 1973.
A. Schrijver,Theory of Linear and Integer Programming, Wiley, New York, 1986.
H. Tverberg, A generalization of Radon's theorem.J. London Math. Soc. 41 (1966), 123–128.
G. Wegner, Über eine kombinatorisch-geometrische Frage von Hadwiger und Debrunner,Israel J. Math. 3 (1965), 187–198.
G. Wegner,d-collapsing and nerves of families of convex sets,Arch. Math. 26 (1975), 317–321.
G. Wegner, Über Helly-Gallaische Stichzahlprobleme, 3,Koll. Discrete Geometry, Salzburg (Institut für Mathematik, Universität Salzburg), 1985, pp. 277–282.
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This research was supported in part by a United States-Israel BSF Grant and by the Fund for Basic Research administered by the Israel Academy of Sciences.
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Alon, N., Kalai, G. Bounding the piercing number. Discrete Comput Geom 13, 245–256 (1995). https://doi.org/10.1007/BF02574042
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DOI: https://doi.org/10.1007/BF02574042